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pogonyaev
3 years ago
15

Find the inverse laplace transform of: (2 s + 4) / (s - 3)^3

Mathematics
1 answer:
Serhud [2]3 years ago
6 0

Answer:

e^{3t}(2t+5t^{2})

Step-by-step explanation:

L^{-1}[\frac{2s+4}{(s-3)^{3}} ]=

Using the Translation theorem to transform the s-3 to s, that means multiplying by and change s to s+3

Translation theorem:L^{1} [F(s-a)=L^{-1}[F(s)|_{s \to s-a}\\ L^{1} [F(s-a)=e^{at} f(t)

L^{-1}[\frac{2s+4}{(s-3)^{3}} ]=e^{3t} L^{-1}[\frac{2(s+3)+4}{s^{3}} ]

Separate the fraction in a sum:

e^{3t} L^{-1}[\frac{2s+10}{s^{3}} ]=e^{3t} L^{-1}[\frac{2s}{s^{3}}+\frac{10}{s^{3}} ]=e^{3t} (L^{-1}[\frac{2}{s^{2}}]+ L^{-1}[\frac{10}{s^{3}}])

The formula for this is:

L^{-1}[\frac{n!}{s^{n+1}} ]=t^{n}

Modify the expression to match the formula.

e^{3t} (2L^{-1}[\frac{1}{s^{1+1}}]+ \frac{10}{2} L^{-1}[\frac{2}{s^{2+1}}])=e^{3t} (2L^{-1}[\frac{1}{s^{1+1}}]+ 5 L^{-1}[\frac{2}{s^{2+1}}])

Solve

e^{3t} (2L^{-1}[\frac{1}{s^{1+1}}]+ 5 L^{-1}[\frac{2}{s^{2+1}}])=e^{3t}(2t+5t^{2} )

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In a random sample of 600 cars making a right turn at a certain intersection, 157 pulled into the wrong lane. Test the hypothesi
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Answer:

a)

<em>The statistic value |Z| = 2.053 > 1.96 at 0.05 Level of significance</em>

<em>Null hypothesis is rejected</em>

<em>Alternative hypothesis is Accepted</em>

<em> Actually 30% of all drivers make not this mistake at the given intersection</em>

<em>b)</em>

<em>The test statistic value |Z| = 2.053 >2.576 at 0.01 Level of significance</em>

<em>Null hypothesis is Accepted</em>

<em> Actually 30% of all drivers make this mistake at the given intersection</em>

Step-by-step explanation:

<u><em>Step(i):</em></u>-

Given Population proportion = 30% or 0.30

Given data In a random sample of 600 cars making a right turn at a certain intersection, 157 pulled into the wrong lane

sample proportion

          p^{-} = \frac{x}{n} = \frac{157}{600} = 0.2616

<em>Null hypothesis:- H₀: p = 0.30</em>

<em>Alternative  hypothesis : H₁:p≠0.30</em>

<u><em>Step(ii):-</em></u>

a)

<em>Test statistic </em>

        Z = \frac{p^{-}-P }{\sqrt{\frac{p(1-p)}{n} } }

      Z = \frac{0.2616-0.30 }{\sqrt{\frac{0.30(1-0.30)}{600} } }

     Z =  -2.053

<em>     |Z| = |-2.053| = 2.053</em>

<em>Level of significance = 0.05</em>

<em>Z₀.₀₅ = 1.96</em>

<em>The calculated value |Z| = 2.053 > 1.96 at 0.05 Level of significance</em>

<em>Null hypothesis is rejected</em>

<em>Alternative hypothesis is Accepted</em>

<u><em>Conclusion:</em></u><em>-</em>

<em> Actually 30% of all drivers make not this mistake at the given intersection</em>

<em>b)</em>

<em>Given level of significance  = 0.01</em>

<em>Z₀.₀₁ = 2.576</em>

<em>The calculated value |Z| = 2.053 >2.576 at 0.01 Level of significance</em>

<em>Null hypothesis is Accepted</em>

<em> Actually 30% of all drivers make this mistake at the given intersection</em>

7 0
2 years ago
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